Ordering results for extreme claim amounts based on random number of claims

Paper: arXiv 2603.24640

Authors: Sangita Das

Abstract

Consider two sequences of heterogeneous and independent portfolios of risks $T_1,T_2,\ldots$ and $T^_{1}, T^{2},\ldots$ and, let $N_1$ and $N_2$ be two positive integer-valued random variables, independent of $T_i’$ and $T^i$, respectively. In this article, we investigate different stochastic inequalities involving $\min{T_1,\ldots,T{N_1}}$ and $\min{T^1,\ldots,T^*{N_2}},$ and $\max{T_1,\ldots,T{N_1}}$ and $\max{T^_1,\ldots,T^_{N_2}}$ in the sense of usual stochastic order and reversed hazard rate order concerning maltivariate chain majorization order. These new results strengthen and generalize some of the well known results in the literature, including \cite{barmalzan2017ordering}, \cite{balakrishnan2018} and \cite{kundu2021_shock} for the case of random claim sizes. Different numerical examples are provided to highlight the applicability of this work. Finally, some interesting applications of our results in reliability theory and auction theory are presented.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents high mathematical complexity focusing on theoretical stochastic orderings and multivariate chain majorization for random claim sizes. However, empirical rigor is minimal as the validation is limited to synthetic numerical examples and theoretical applications rather than real-world quantitative finance backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Compare extreme claim amounts <br>from random-sized risk portfolios] --> B[Inputs: Independent risk sequences <br>T_i, T*_i and claim counts N1, N2]
    B --> C[Methodology: Apply Multivariate Chain <br>Majorization & Stochastic Orders]
    C --> D[Analysis: Compare Min/Max claim sizes <br>via Usual Stochastic & Reversed Hazard Rate Orders]
    D --> E[Validation: Execute numerical examples <br>to confirm theoretical bounds]
    E --> F[Outcomes: Generalized prior models + <br>Applications in Reliability & Auction Theory]

Cited References (347)

1. Some ordering results for the M arshall and O lkin’s family of distributions (2019) — link
2. Ordering properties of the smallest order statistic from W eibull G random variables (2024) — link
3. Ordering results between extreme order statistics in models with dependence defined by Archimedean [survival] copulas (2022)
4. Ordering results for smallest claim amounts from two portfolios of risks with dependent heterogeneous exponentiated location-scale claims (2022) — link
5. Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples (2020) — [link](https://doi.org/10.1007/ s00184-019-00753-2)
6. A F irst C ourse in O rder S tatistics (1992)
7. Permanents, order statistics, outliers, and robustness (2007) — link
8. Comparisons between largest order statistics from multiple-outlier models (2016) — link

0 of 347 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.